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Mosc. Math. J., 2018, Volume 18, Number 2, Pages 367–386 (Mi mmj676)  

Bounding the length of iterated integrals of the first nonzero Melnikov function

Pavao Mardešića, Dmitry Novikovb, Laura Ortiz-Bobadillac, Jessie Pontigo-Herrerab

a Université de Bourgogne, Institute de Mathématiques de Bourgogne — UMR 5584 CNRS, Université de Bourgogne, 9 avenue Alain Savary, BP 47870, 21078 Dijon, FRANCE
b Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, 7610001, Israel
c Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de la Investigación Científica, Circuito exterior, Ciudad Universitaria, 04510, Ciudad de México, México

Abstract: We consider small polynomial deformations of integrable systems of the form $dF=0$, $F\in\mathbb C[x,y]$ and the first nonzero term $M_\mu$ of the displacement function $\Delta(t,\epsilon)=\sum_{i=\mu}M_i(t)\epsilon^i$ along a cycle $\gamma(t)\in F^{-1}(t)$. It is known that $M_\mu$ is an iterated integral of length at most $\mu$. The bound $\mu$ depends on the deformation of $dF$.
In this paper we give a universal bound for the length of the iterated integral expressing the first nonzero term $M_\mu$ depending only on the geometry of the unperturbed system $dF=0$. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for $M_\mu$ to be given by an abelian integral, i.e., by an iterated integral of length $1$. We conjecture that our bound is optimal.

Key words and phrases: Hilbert 16th problem, center problem, Poincaré return map, abelian integrals, limit cycles, free group automorphism.

Full text: http://www.mathjournals.org/.../2018-018-002-007.html
References: PDF file   HTML file

Document Type: Article
MSC: Primary 34C07; Secondary 34C05, 34C08
Language: English

Citation: Pavao Mardešić, Dmitry Novikov, Laura Ortiz-Bobadilla, Jessie Pontigo-Herrera, “Bounding the length of iterated integrals of the first nonzero Melnikov function”, Mosc. Math. J., 18:2 (2018), 367–386

Citation in format AMSBIB
\Bibitem{MarNovOrt18}
\by Pavao~Marde{\v s}i\'c, Dmitry~Novikov, Laura~Ortiz-Bobadilla, Jessie~Pontigo-Herrera
\paper Bounding the length of iterated integrals of the first nonzero Melnikov function
\jour Mosc. Math.~J.
\yr 2018
\vol 18
\issue 2
\pages 367--386
\mathnet{http://mi.mathnet.ru/mmj676}


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